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    "## [线性回归从零开始实现-参考链接](https://zh-v2.d2l.ai/chapter_linear-networks/linear-regression-scratch.html)\n",
    "- [1.生成数据](#1)\n",
    "- [2.读取数据集](#2)\n",
    "- [3.初始化参数模型](#3)\n",
    "- [4.定义模型](#4)\n",
    "- [5.损失函数](#5)\n",
    "- [6.优化算法](#6)\n",
    "- [7.训练](#7)\n",
    "\n",
    "## [线性回归简洁实现](#8)"
   ]
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   "source": [
    "### <span id = '1'> 1. 生成数据</span>"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ead8ea49",
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   "source": [
    "import random\n",
    "import torch\n",
    "from d2l import torch as d2l\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e73dd101",
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   "source": [
    "&emsp;在下面的代码中，我们生成一个包含1000个样本的数据集，每个样本包含从标准正态分布中采样的2个特征。我们的合成数据集是一个矩阵 $\\mathbf{X}\\in \\mathbb{R}^{1000 \\times 2}$。<br>\n",
    "&emsp;我们使用线性模型参数$\\mathbf{w} = [2, -3.4]^\\top$、$b = 4.2$和噪声项$\\epsilon$生成数据集及其标签：\n",
    "\n",
    "$$\\mathbf{y}= \\mathbf{X} \\mathbf{w} + b + \\mathbf\\epsilon.$$\n",
    "\n",
    "&emsp;你可以将 $\\epsilon$ 视为捕获特征和标签时的潜在观测误差。在这里我们认为标准假设成立，即$\\epsilon$服从均值为0的正态分布。\n",
    "为了简化问题，我们将标准差设为0.01。"
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a16da6cd",
   "metadata": {},
   "outputs": [],
   "source": [
    "def synthetic_data(w, b, num_examples):\n",
    "    X = torch.normal(0, 1, (num_examples, len(w)))  # shape:(num_examples, len(w))\n",
    "    y = torch.matmul(X, w) + b\n",
    "    y += torch.normal(0, 0.01, y.shape)  # 误差e\n",
    "    return X, y.reshape((-1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "02fd78bc",
   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "feature:  tensor([-0.6019, -1.0271]) \n",
      "label:  tensor([6.4807])\n"
     ]
    }
   ],
   "source": [
    "true_w = torch.tensor([2, -3.4])\n",
    "true_b = 4.2\n",
    "features, labels = synthetic_data(true_w, true_b, 1000)\n",
    "\n",
    "print('feature: ', features[0], '\\nlabel: ', labels[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a832a399",
   "metadata": {},
   "source": [
    "`features` 中的每一行都包含一个二维数据样本,shape:(1000,2)<br>\n",
    "`labels` 中的每一行都包含一维标签值（一个标量）,shape:(1000,1) <br><br>\n",
    "通过生成第二个特征 `features[:, 1]` 和 `labels` 的散点图，可以直观地观察到两者之间的线性关系"
   ]
  },
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   "id": "de155f67",
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      "text/plain": [
       "<Figure size 252x180 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "d2l.set_figsize()\n",
    "d2l.plt.scatter(features[:, 1].detach().numpy(),\n",
    "               labels.detach(), 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "fefe53c7",
   "metadata": {},
   "outputs": [],
   "source": [
    "# d2l.set_figsize()\n",
    "# d2l.plt.scatter(features[:, 0].detach().numpy(),  \n",
    "#                labels.detach(), 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4386be01",
   "metadata": {},
   "source": [
    " ### <span id='2'> 2. 读取数据集</span>\n",
    " - 定义一个 `data_iter` 函数， 该函数接收批量大小、特征矩阵和标签向量作为输入，生成大小为 `batch_size` 的小批量。每个小批量包含一组特征和标签。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5c1e4d4f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    \n",
    "    random.shuffle(indices)\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        batch_indices = torch.tensor(indices[i:min(i+batch_size, \n",
    "                                                  num_examples)])\n",
    "        yield features[batch_indices], labels[batch_indices]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b4c27da7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[-0.3873,  1.0235],\n",
      "        [-0.6841,  0.0206],\n",
      "        [ 0.5101,  1.1249],\n",
      "        [ 1.6261, -1.8431],\n",
      "        [-1.3831, -0.1604],\n",
      "        [-1.7686,  0.6447],\n",
      "        [-0.1321,  0.5082],\n",
      "        [ 0.0683, -0.9338],\n",
      "        [ 1.4336,  0.2267],\n",
      "        [ 2.2143,  0.7623]]) \n",
      " tensor([[-0.0516],\n",
      "        [ 2.7595],\n",
      "        [ 1.3990],\n",
      "        [13.7160],\n",
      "        [ 1.9771],\n",
      "        [-1.5413],\n",
      "        [ 2.2000],\n",
      "        [ 7.5213],\n",
      "        [ 6.3025],\n",
      "        [ 6.0387]])\n"
     ]
    }
   ],
   "source": [
    "batch_size = 10\n",
    "\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X, '\\n', y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "959f7a4c",
   "metadata": {},
   "source": [
    "### <span id='3'>3. 初始化参数模型</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6e7a48dd",
   "metadata": {},
   "outputs": [],
   "source": [
    "w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)\n",
    "b = torch.zeros(1, requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "447f4e86",
   "metadata": {},
   "source": [
    "### <span id='4'> 4. 定义模型</span>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "eda406cd",
   "metadata": {},
   "outputs": [],
   "source": [
    "def linreg(X, w, b):\n",
    "    return torch.matmul(X,w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb7fec5f",
   "metadata": {},
   "source": [
    "### <span id='5'> 5.损失函数</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "cf48024c",
   "metadata": {},
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):\n",
    "    return (y_hat - y.reshape(y_hat.shape))**2 /2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d112671",
   "metadata": {},
   "source": [
    "### <span id='6'> 6.优化算法</span>\n",
    "- 小批量随机梯度下降\n",
    "$$\\qquad\\quad\\theta_{j}:=\\theta_{j}+ \\frac{\\alpha}{batch-num} \\sum_{i=1}^{batch-num}(y^{(i)}-h_{\\theta}(x^{(i)}))x_j^{(i)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "91db238c",
   "metadata": {},
   "outputs": [],
   "source": [
    "def sgd(params, lr, batch_size):\n",
    "    with torch.no_grad():\n",
    "        for param in params:\n",
    "            param -= lr * param.grad / batch_size\n",
    "            param.grad.zero_()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a831fb90",
   "metadata": {},
   "source": [
    "### <span id='7'> 7.训练</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "81e622f0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.033933\n",
      "epoch 2, loss 0.000123\n",
      "epoch 3, loss 0.000053\n"
     ]
    }
   ],
   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "net = linreg\n",
    "loss = squared_loss\n",
    "\n",
    "for epoch in range(num_epochs):\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X, w, b), y)  # 'X' 和 'y' 是小批量损失\n",
    "        # 因为‘l’形状是（batch_size, 1），而不是一个标量。‘l’中所有元素被加到一起\n",
    "        # 并以此计算关于['w', 'b']的梯度\n",
    "        l.sum().backward()\n",
    "        sgd([w, b], lr, batch_size)  # 使用参数的梯度更新参数\n",
    "    \n",
    "    with torch.no_grad():\n",
    "        train_l = loss(net(features, w, b), labels)\n",
    "        print(f'epoch {epoch+1}, loss {float(train_l.mean()):f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "fcb5d46e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w的估计误差：tensor([-4.8518e-04, -3.6478e-05], grad_fn=<SubBackward0>)\n",
      "b的估计误差：tensor([-0.0002], grad_fn=<RsubBackward1>)\n"
     ]
    }
   ],
   "source": [
    "print(f'w的估计误差：{true_w - w.reshape(true_w.shape)}')\n",
    "print(f'b的估计误差：{true_b - b}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab6198af",
   "metadata": {},
   "source": [
    "# <span id='8'>线性回归简洁实现</span>\n",
    "[参考链接](https://zh-v2.d2l.ai/chapter_linear-networks/linear-regression-concise.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4ceec3c",
   "metadata": {},
   "source": [
    "### 生成数据集"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8f82d9da",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import torch\n",
    "from torch.utils import data\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "b456226a",
   "metadata": {},
   "outputs": [],
   "source": [
    "true_w2 = torch.tensor([2, -3.4])\n",
    "true_b2 = 4.2\n",
    "features2, labels2 = d2l.synthetic_data(true_w2, true_b2, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a60835f9",
   "metadata": {},
   "source": [
    "### 读取数据\n",
    "- 实例化数据迭代器对象时指定 batch_size\n",
    "- 布尔值 is_train 表示是否希望数据迭代器对象在每个迭代周期内打乱数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "fe47158b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def load_array(data_arrays, batch_size, is_train=True):\n",
    "    dataset = data.TensorDataset(*data_arrays)\n",
    "    return data.DataLoader(dataset, batch_size, shuffle=is_train)\n",
    "\n",
    "batch_size2 = 10\n",
    "data_iter2 = load_array((features2, labels2), batch_size2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "1881c8fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[tensor([[ 0.3798, -0.2506],\n",
       "         [-1.4702,  0.9068],\n",
       "         [-1.3497, -1.0018],\n",
       "         [-1.2756,  0.1085],\n",
       "         [ 0.6012,  0.1379],\n",
       "         [ 1.0505,  1.4996],\n",
       "         [ 0.4775,  1.2668],\n",
       "         [-1.3526, -0.4193],\n",
       "         [-1.6692,  0.6879],\n",
       "         [ 0.6611, -0.9381]]),\n",
       " tensor([[ 5.8064],\n",
       "         [-1.8342],\n",
       "         [ 4.9042],\n",
       "         [ 1.2693],\n",
       "         [ 4.9262],\n",
       "         [ 1.1954],\n",
       "         [ 0.8257],\n",
       "         [ 2.9330],\n",
       "         [-1.4576],\n",
       "         [ 8.7207]])]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "next(iter(data_iter2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94a8a4d2",
   "metadata": {},
   "source": [
    "### 定义模型\n",
    "- 全连接层在 Linear 类中定义。值得注意的是，我们将两个参数传递到 nn.Linear 中。第一个指定输入特征形状，即 2，第二个指定输出特征形状，输出特征形状为单个标量，因此为 1。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "d7cc5991",
   "metadata": {},
   "outputs": [],
   "source": [
    "from torch import nn\n",
    "net = nn.Sequential(nn.Linear(2, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe29f77c",
   "metadata": {},
   "source": [
    "### 初始化参数\n",
    "- 正如我们在构造 nn.Linear 时指定输入和输出尺寸一样。现在我们直接访问参数以设定初始值。我们通过 net[0] 选择网络中的第一个图层，然后使用 weight.data 和 bias.data 方法访问参数。然后使用替换方法 normal_ 和 fill_ 来重写参数值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "1cd953f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0.])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "net[0].weight.data.normal_(0, 0.01)\n",
    "net[0].bias.data.fill_(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "299d44bd",
   "metadata": {},
   "source": [
    "### 定义损失函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "9b8b804f",
   "metadata": {},
   "outputs": [],
   "source": [
    "loss = nn.MSELoss()  # 均方误差"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b7f3ed6",
   "metadata": {},
   "source": [
    "### 优化算法\n",
    "- 当我们实例化 SGD 实例时，我们要指定优化的参数（可通过 net.parameters() 从我们的模型中获得）以及优化算法所需的超参数字典。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "02563701",
   "metadata": {},
   "outputs": [],
   "source": [
    "trainer = torch.optim.SGD(net.parameters(), lr=0.03)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd26f7c2",
   "metadata": {},
   "source": [
    "### 训练\n",
    "在每个迭代周期里，我们将完整遍历一次数据集（train_data），不停地从中获取一个小批量的输入和相应的标签。对于每一个小批量，我们会进行以下步骤:\n",
    "\n",
    "- 通过调用 net(X) 生成预测并计算损失 l（正向传播）。\n",
    "\n",
    "- 通过进行反向传播来计算梯度。\n",
    "\n",
    "- 通过调用优化器来更新模型参数。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "76e7da3e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.000201\n",
      "epoch 2, loss 0.000097\n",
      "epoch 3, loss 0.000097\n"
     ]
    }
   ],
   "source": [
    "num_epochs = 3\n",
    "for epoch in range(num_epochs):\n",
    "    for X, y in data_iter2:\n",
    "        l = loss(net(X), y)\n",
    "        trainer.zero_grad()\n",
    "        l.backward()\n",
    "        trainer.step()\n",
    "    l = loss(net(features2), labels2)\n",
    "    print(f'epoch {epoch + 1}, loss {l:f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "af503b1d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w的估计误差： tensor([-0.0003, -0.0007])\n",
      "b的估计误差： tensor([8.5354e-05])\n"
     ]
    }
   ],
   "source": [
    "w = net[0].weight.data\n",
    "print('w的估计误差：', true_w2 - w.reshape(true_w.shape))\n",
    "b = net[0].bias.data\n",
    "print('b的估计误差：', true_b2 - b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "288bebae",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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